## Ohio State University Algebraic Geometry Seminar## Year 2017-2018Time: Tuesdays 3-4pmLocation: MW 154 |
---|

(Anderson): Relations between degeneracy loci and divisors on curves go back over a hundred years: Giambelli's formula for the class of the locus where a matrix drops rank is a cornerstone of Schubert calculus; Brill and Noether's estimate for the dimension of the space of special divisors on a curve also comes from considerations of degeneracy loci. In this talk, I will describe work with Linda Chen and Nicola Tarasca which connects more recent developments in Schubert calculus with formulas by Eisenbud-Harris, Pirola, and Chan-Lopez-Pflueger-Teixidor for the genus of a Brill-Noether curve.

(Tseng):
The quantum K-ring of a smooth projective variety X, introduced by Givental and YP Lee, is a deformation of the Grothendieck ring of coherent sheaves on X defined using holomorphic Euler characteristics on moduli spaces of stable maps to X (these are K-theoretic version of Gromov-Witten invariants). In this talk we discuss some properties of the quantum K-rings of X=Fl_{r+1}, the variety of complete flags in C^{r+1}, including:

(1) a presentation of the quantum K-ring;

(2) finiteness of quantum product;

(3) canonical polynomial representatives of Schubert classes.

This is based on joint work in progress with David Anderson and Linda Chen.

(Cueto): In this talk, I will discuss the structure of tropical and non-Archimedean analytic genus 2 curves and their moduli from three perspectives:

(1) as 2-to-1 covers of **P ^{1}** branched at 6 points;

(2) as solutions to the hyperelliptic equation y^{2}=f(x), where f has degree 5; and

(3) as metric graphs dual to genus 2 nodal algebraic curves over a valued field.

Our first description allows us to give an explicit combinatorial rule to characterize each such
metric graph together with a harmonic map to a metric tree on 6 leaves, in terms of the valuations of 6 branch
points in **P ^{1}**. Even though the tropicalization of a plane hyperelliptic curve shows no genus, we provide
explicit re-embeddings of the input planar hyperelliptic curve that reveals the correct metric graphs.

From our third viewpoint, we consider the moduli space of abstract genus two tropical curves and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm. While these tropical Igusa functions do not yield coordinates on the tropical moduli space, we propose an alternative set of invariants that provides new length data. This is joint work with Hannah Markwig.

(Clemens): The talk will construct one example of the `equivalence' between an elliptically-fibered CY-threefold endowed with a semi-stable E_{8} + E_{8} bundle and a semi-stable degeneration of an elliptically-fibered CY-fourfold. I will then loosely describe how this equivalence is used to `break' the SU(5)-symmetry of grand unified theory (GUT) to the SU(3)xSU(2)xU(1)-symmetry of what physicists call the Standard Model of the physics of our cosmos.

(Baldwin): The cohomology ring of flag varieties has long been known to exhibit positivity properties. One such property is that the structure constants of the Schubert basis with respect to the cup product are non-negative. Brion (2002) and Anderson-Griffeth-Miller (2011) have shown that positivity extends to K-theory and T-equivariant K-theory, respectively. In this talk I will discuss recent work (joint with Shrawan Kumar) which generalizes these results to the case of Kac-Moody groups.

(Docampo): We study the sheaf of Kähler differentials on the arc space of an algebraic variety. We obtain explicit formulas that can be used effectively to understand the local structure of the arc space. The approach leads to new results as well as simpler and more direct proofs of some of the fundamental theorems in the literature. The main applications include: an interpretation of Mather discrepancies as embedding dimensions of certain points in the arc space, a new proof of a version of the birational transformation rule in motivic integration, and a new proof of the curve selection lemma for arc spaces. This is joint work with Tommaso de Fernex.

(Bibby): We are interested in certain arrangements of subvarieties on which a wreath product group acts. We give a combinatorial description of its poset of layers (connected components of intersections) as a generalization of Dowling and partition lattices. This combinatorial structure is an aid in understanding the cohomology of the complement as a representation of the group. In particular, when one considers a sequence of these representations, its stability can be viewed as a consequence of combinatorial stability. Joint work with Nir Gadish.

(Manon): I'll give an introduction with examples to the notion of Khovanskii basis. Khovanskii bases are generating sets for algebras which are distinguished by their computational properties as well as their relationships with tropical geometry and toric geometry. After a few examples, I'll survey some recent results on their existence and classification.

(Litt): Let *X* be an algebraic variety over a field *k*. Which representations of π_{1}(*X*) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over *X*? We study this question by analyzing the action of the Galois group of *k* on the fundamental group of *X*.
As a sample application of our techniques, we show that if *X* is a normal variety over a field of characteristic zero, and *p* is a prime, then there exists an integer N=N(*X*,*p*) satisfying the following: any irreducible, non-trivial *p*-adic representation of the fundamental group of *X*, which arises from geometry, is non-trivial mod *p*^{N}.

(Gonzalez): The Cox rings of weighted projective planes blown-up at a general point have been studied in connection to the noetherianity of symbolic Rees algebras and to the finite generation of the Cox ring of the moduli space of stable rational pointed curves. We will present some sufficient conditions for the finite and nonfinite generation of the Cox ring of such blowups. This is joint work with Javier Gonzalez and Kalle Karu.

(Khan): One of the central questions in complex geometry is to understand the moduli space of complex structures on a given manifold. In general, this is a mysterious object. Locally, we can study deformations of complex structures using Kodaira-Spencer theory. However, the global geometry can be very complicated and have singularities. In this talk, we consider the case where we have a Riemannian-flat metric and show how this can be used to understand the geometry of the moduli space. From here, we lay out some conjectures about how this can be done more generally, and give some heuristic evidence for these conjectures.

**Ohio State University Algebraic Geometry Seminar-Year 2016-2017**

**Ohio State University Algebraic Geometry Seminar-Year 2015-2016**

**Ohio State University Algebraic Geometry Seminar-Year 2014-2015**

**Ohio State University Algebraic Geometry Seminar-Year 2013-2014**

This page is maintained by Angie Cueto and Dave Anderson.